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My Bridge Hammock System - Part 7

Setting Cl to the compression force due to the suspension lines, we have:

Code:

Cl = Lf/(2 * tan(beta)) = Wh/[4*d*tan(beta)]

The next figure, #5, illustrates the suspension line plane including the forces acting on the spreader bar through the webbing.

In order to obtain the angle gamma, we must measure the angle of the tangent line to the end arc and the perpendicular to the spreader bar. To obtain a good estimate of this, measure down the length of the webbing from the spreader bar a short distance (a short distance is desired since if the distance is too great, the angle will diverge more from the tangent to the webbing/spectra at the spreader bar) Em as indicated in the next figure #6.

At that point on both ends, measure the distance between the end points of the hammock, Hw. Note Hw is NOT the length of the hammock, i.e., the length of the fabric between the two points. Also, make sure to measure the distance while you or someone is in the hammock.

We have:

Code:

sine(gamma) = (S-Hw)/2 * Em

Then setting Cw to the compression due to the weight of the occupant, we have:

I have included below a program in the "Awk Programming Language" for computing the compressive forces on spreader bars and for determining the parabola needed for draft stoppers and bug netting end panels.

To execute in Linux, copy to a file, make the file executable and name the file on command line. If you run Linux, you probably already know how to do this.

I know zero about Windows and even less desire to know anything about it and less about Macs, so you will have to get somebody else to help you there if you want to run the program. I am pretty sure that Awk is available, but don't know where. There are probably people on the forums that know a lot about both Windows and the Mac and can undoubtedly help you.

Modify the values in the "User Input Section at the top and run the program using Awk.

You can detail the values in either English units or metric units or a combination of both. If the same measurement is detailed in both English and metric, the English value will be used.

I have detailed the compression forces above.

The derivation of the parabolic arc length is included here for those so inclined. Warning: if you do not like math, then skip this section. Also, for those who are advanced in mathematics, this will be very elementary. I have included all of this here for completeness so that those so inclined can read, adapt and improve.

For the draft stopper we have two data points for the parabola:

the spreader bar width, and

the width of the fabric at the end, i.e., the arc length of the

needed parabola.

A parabola is a second degree equation. To make things easy I will assume the co-ordinate system origin at the vertex of the parabola with the parabola opening up the positive y-axis. This reduces the needed equation to:

Code:

x^2 = 2py ('^' denotes exponential)

The equation constant, p, needs to be defined to define the parabola. I will use the fabric width to define the p. For this I need the equation for the arc length of a parabola - it's been too many years for me to even try to remember that piece of information. So, arc length, L, is the integral of the equation along the arc: NOTE - there is a fancy math symbol for the integral, but computers haven't really caught up with several hundred years of math yet and so you have to do some fancy things to get everybody to properly display simple math symbols. So I will simply write "integral" for the symbol.

Code:

L = integral{sqrt[1 + (dy/dx)^2]dx}

"sqrt" == square root - again another math symbol, mathematicians are also really good at inventing symbols.

from the parabolic equation:

Code:

dy/dx = x/p

So we want:

Code:

L = integral{sqrt[1 + (x/p)^2]dx}

Now I learned over 44 years ago, that at this point I pull out my trusty copy of "CRC Standard Mathematical Tables, 25th Edition", 1978 - yes I'm old enough that I bought it brand new after more than a decade of hungering for a copy. Looking up the table of indefinite integrals, I find one that fits the bill, namely

Unfortunately this cannot be solved explicitly for p. So, what we have to do is insert a trial value for p and compute the corresponding value for L. If L is not equal to the desired fabric width, we keep trying values for p until we get a value of L that is acceptably close to the desired value. Of course, we don't try values of p randomly, but try to do so systematically. Now this kind of computation is what digital computers are very, very, very good at doing. That is exactly what the following program does. It starts with a reasonable guess for p and then keeps trying values until L is within a specified tolerance of the desired value for L.

So, once we have p, we can now compute the data points, i.e., x and y pairs, for the desired parabola. The program then proceeds to do so and prints out information that you have specified and the data points for the parabola.

I have arranged the program so that you are not flooded with lots and lots of data points to plot. I found that the first 5 data points along the y-axis, i.e., the first 5 cm, are needed and after that one every 5 cm will do. That will give approximately 12 to 15 points for the end of a Bridge Hammock. You don't even need to get fancy in connecting the data points, simple straight lines between each is accurate enough for this use. Simple.

All internal calculations are done in metric units.

Note: some Awks do not recognize the inverse trig function 'acos' or arc-cosine. For quite a few years now, I have been using an implementation of Awk which I developed and wrote. It runs under Linux since that is all I use. I have extended Awk considerably, especially in the mathematical functions since the original Awk is weak there and in the ability to dynamically load/unload and run compiled functions written in C or some other compiled language. Thus, I tend to use functions that are not available under normal Awk. Sorry about that, but maybe somebody can work around the issue. The 'acos' function is only used for the compression force calculations and may be commented out to run the rest under normal Awk. If all else fails maybe somebody can compile under C since my implementation of Awk is tailered to look almost exactly like C. I was going to write a C interpreter, but decided instead that Awk was a better language for an interpreter.

Note 2: The variable 'sanity_check' should be set to zero, 0, for ordinary use. The variable was put in to add a check on the computations to ensure agreement with the integrated value for the arc length. With the value of 'sanity_check' set to 1, the straight distances between computed parabolic points is computed and accumulated. The total should approach the integrated value as the distance between computed points is decreased.